Partial Fraction Decomposition Calculator

Your input: perform the partial fraction decomposition of $$$\frac{x}{\left(x - 5\right) \left(x - 3\right)}$$$

The form of the partial fraction decomposition is

$$\frac{x}{\left(x - 5\right) \left(x - 3\right)}=\frac{A}{x - 5}+\frac{B}{x - 3}$$

Write the right-hand side as a single fraction:

$$\frac{x}{\left(x - 5\right) \left(x - 3\right)}=\frac{\left(x - 5\right) B + \left(x - 3\right) A}{\left(x - 5\right) \left(x - 3\right)}$$

The denominators are equal, so we require the equality of the numerators:

$$x=\left(x - 5\right) B + \left(x - 3\right) A$$

Expand the right-hand side:

$$x=x A + x B - 3 A - 5 B$$

Collect up the like terms:

$$x=x \left(A + B\right) - 3 A - 5 B$$

The coefficients near the like terms should be equal, so the following system is obtained:

$$\begin{cases} A + B = 1\\- 3 A - 5 B = 0 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=\frac{5}{2}$$$, $$$B=- \frac{3}{2}$$$

Therefore,

$$\frac{x}{\left(x - 5\right) \left(x - 3\right)}=\frac{\frac{5}{2}}{x - 5}+\frac{- \frac{3}{2}}{x - 3}$$

Answer: $$$\frac{x}{\left(x - 5\right) \left(x - 3\right)}=\frac{\frac{5}{2}}{x - 5}+\frac{- \frac{3}{2}}{x - 3}$$$